Partial fraction expansions and zeros of Hankel transforms

It is proved by the method of partial fraction expansions and Sturm’s oscillation theory that the zeros of certain Hankel transforms are all real and distributed regularly between consecutive zeros of Bessel functions. As an application, the sufficient or necessary conditions on parameters for which ${}_1{F}_2$ hypergeometric functions belong to the Laguerre-Pólya class are investigated in a constructive manner.